A polynomial variant of diophantine triples in linear recurrences

Abstract

Let (G_n)_{n=0}^{infty } be a polynomial power sum, i.e. a simple linear recurrence sequence of complex polynomials with power sum representation G_n = f_1alpha _1^n + cdots + f_kalpha _k^n and polynomial characteristic roots alpha _1,ldots ,alpha _k . For a fixed polynomial p, we consider sets left{ a,b,c right} consisting of three non-zero polynomials such that ab+p, ac+p, bc+p are elements of (G_n)_{n=0}^{infty } . We will prove that under a suitable dominant root condition there are only finitely many such sets if neither f_1 nor f_1 alpha _1 is a perfect square.